Spinor Info

It looks like the Mythbusters tend to ignore air resistance.

In a recent episode, they claimed to have demonstrated that a horizontally fired bullet and a bullet that is simply dropped fall to the ground in the same amount of time. They were wrong. What they actually demonstrated is that air resistance causes the fired bullet to hit the ground more slowly.

Their argument would apply perfectly in a vacuum, as on the surface of the Moon, but not here on the Earth, where the bullet’s motion is governed not just by the laws of gravity, but also by the laws of a non-conservative force, namely air resistance. (Why is it non-conservative? Some of the bullet’s kinetic energy is converted into heat, as it travels through the air at high speed. Unless we also include the thermodynamics of the air into our equations of motion, the equations will not conserve energy, as the amount of kinetic energy converted into heat will just appear “lost”.)

The bullet’s velocity, v, can be written as v² = v_h² + v_v², where v_h is the horizontal and v_v is the vertical component. The initial horizontal velocity is v₀. The initial vertical velocity is 0.

Air resistance is proportional to the square of the bullet’s velocity. To be precise, acceleration due to air resistance will be

a_air = κv²,

where κ is an unknown proportionality factor. (To be more precise, κ = ½cAρm, where c is the dimensionless drag coefficient, A is the bullet’s cross-sectional area, ρ is the density of the air, and m is the bullet’s mass.) The direction of the acceleration will be opposite the direction of the bullet’s motion.

The total acceleration of the bullet will have two components: a vertical component g due to gravity, and a component a_air opposite the direction of the bullet’s motion.

The resulting equations of motion can be written as:

dv_h/dt = –κvv_h,
dv_v/dt = g – κvv_v.

Right here we can see the culprit: air resistance not only slows the bullet down horizontally, it also reduces its downward acceleration.

This is a simple system of two differential equations in the two unknown functions v_h(t) and v_v(t). Its solution is not that simple, unfortunately. However, it can be greatly simplified if we notice that given that v_v << v_h, v ≅ v_h, and therefore, we get

dv_h/dt = –κv_h²,
dv_v/dt = g – κv_vv_h.

This system is solved by

v_h = 1/(κt + C₁),
v_v = v_h[(½κt² + C₁t)g + C₂].

Given

v₀ = 1/(κt₀ + C₁)

we have

C₁ = 1/(v₀ – κt₀),

which leads to

v_h = v₀/[κ(t – t₀)v₀ + 1],

or, if we set t₀ = 0,

v_h = v₀/(κtv₀ + 1),

Similarly, given v_v(0) = 0, we get C₂ = 0, thus

v_v = gt(κv₀t + 2)/(2κv₀t + 2).

The distance traveled horizontally (s_h) and vertically (s_v) between t₀ = 0 and t₁ can be obtained by simple integration of the respective velocities with respect to t between 0 and t₁:

s_h = log(κv₀t₁ + 1)κ,
s_v = g[κv₀t₁(κv₀t₁ + 2) – 2log(κv₀t₁ + 1)]/(2κv₀)².

The claim by the Mythbusters was that the time it took for the fired bullet to hit the ground was only ~40 ms more than the time it took for a dropped bullet to fall, which is a negligible difference. But it is not! Taking g ≅ 10 m/s², it is easy to see that the time it takes for a bullet to fall from a height of 1 m, using the well-known formula ½gt², is 447 ms; the difference measured by the Mythbusters is nearly 10% of this number!

Not only did the fired bullet take longer to hit the ground, the Myhtbusters’ exquisite setup allows us to calculate the bullet’s initial velocity v₀ and drag coefficient κ. This is possible because the Mythbusters conveniently provided three pieces of information (I am using approximate numbers here): the length of the path that the bullet traveled s_h ≅ 100 m), the height of the bullet at the time of firing (s_v ≅ 1 m), and the time it took for the fired bullet to hit the ground. Actually, what they provided was the difference between the time for a fired vs. a dropped bullet to hit the ground, but we know what it is for the dropped bullet (and because it is never moving very rapidly, we can ignore air resistance in its case), so t₁ = 447 + 40 = 487 ms. The solution is given by

κ = 0.0054 m^–1,
v₀ = 272.3 m/s.

Given a bullet cross-sectional area of A = 2 cm² = 2 × 10^–4 m², an approximate air density of ρ = 1 kg/m³, and a bullet mass of m = 20 g = 0.02 kg, the dimensionless drag coefficient for the bullet can be calculated as c = 2κmAρ = 1.08, which is not at all unreasonable for a tumbling bullet. Of course the actual values of A and m may differ from the ones I’m using here, resulting in a different value for the dimensionless drag coefficient c.

First, it was the multiverse. Then came Boltzmann brains. Now here’s another intriguing idea: the cancellation of the Superconducting Supercollider project in the 1990s and last year’s failure of one of the Large Hadron Collider’s magnets at CERN are just two manifestations of manifest bad luck brought about by the fact that the Higgs particle simply cannot be discovered; that its discovery at any time in the future propagates backwards in time, causing events that prevent its discovery in the first place.

An intriguing idea, though not precisely original, as something much like it was already published in the form of John Cramer‘s excellent science fiction novel, Einstein’s Bridge. And when I say intriguing, I mean intriguing… as the basis of a science fiction story. But as the basis of a scientific paper? Another adjective comes to my mind… appalling.

I recently read a review of Weinberg’s wonderful new book, Cosmology, a 2009 sequel of sorts to his 1972 classic, Gravitation and Cosmology. The reviewer mentioned two other books, that of Mukhanov and that of Dodelson, as books worth having. Mukhanov’s Physical Foundations of Cosmology was already on my bookshelf (and, like the reviewer, I also consider it worth having) but not Dodelson’s book… so I decided to buy it.

I was not disappointed: it is an excellent cosmology book. In particular, it offers a very thorough introduction to the quantitative aspects of physical cosmology.

However… although the book was published only six years ago, it feels surprisingly dated. Through no fault of the author, to be clear: it’s just that cosmology has made tremendous progress in a few short years. I can think of two things in particular: results from the Wilkinson Microwave Anisotropy Probe (WMAP), first released in 2005, providing precision maps of the cosmic microwave background, allowing accurate detection of the so-called acoustic peaks; and ever improving large scale galaxy surveys, notably the Sloan Digital Sky Survey (SDSS), providing spectra for many hundreds of thousands of galaxies, yielding 3D density maps of the deep cosmos that can be used to test models of structure formation.

The results speak for themselves. For instance, Dodelson’s book gives 12.6 ± 1.1 billion years as the age of the Universe… in contrast, the latest WMAP result is 13.73 ± 0.12 billion years, a tenfold improvement in the accuracy of the estimate. I guess it’s not an enviable task to write a book for a field that is changing as rapidly as Modern Cosmology… which also happens to be the title of Dodelson’s book.